Titu_Andreescu,_Bogdan_Enescu_Cauchy_Schwarz_Inequality_Yet_Another

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Cauchy-Schwarz Inequality: Yet Another Proof
Titu Andreescu and Bogdan Enescu give an elegant — and memorable —
proof of the Cauchy-Schwarz inequality among the gems in their Mathematical
Olympiad Treasures (Birkhauser, 2003). Nominally, the proof is inductive, but
what I like so much about it is that the induction step comes as close to being
“computation free” as one can imagine.
The proof begins with a simple lemma that is useful in its own right.
Lemma: For real a and b and for x > 0, and y > 0, one has
(a + b)2
x + y
≤ a
2
x
+
b2
y
. (1)
Naturally, this lemma is trivial — once it is conceived. Just by expansion
and factorization one finds that it is simply a restatement of (ay − bx)2 ≥ 0.
What I find attractive about this inequality is the way that it is self-generalizing.
For example, if we replace b by b + c and replace y by y + z the we find
(a + b + c)2
x + y + z
≤ a
2
x
+
(b + c)2
y + z
≤ a
2
x
+
b2
y
+
c2
z
. (2)
Repeating this step then gives us the general relationship
(a1 + a2 + · · ·+ an)2
x1 + x2 + · · ·+ xn
≤ a
2
1
x1
+
a22
x2
+ · · ·+ a
2
n
xn
, (3)
from which we can see Cauchy-Schwarz inequality at a glance. One simply sets
ak = αkβk and xk = β2k.
How Different Really?
The related positivity bound (a − bx)2 ≥ 0 is a familiar starting place for
proofs of the Cauchy-Schwarz inequality, so one can ask if this proof is really
so different. The answer depends on where an individual chooses to draw some
mental boundaries. To me, the appearance of the fractional bound (1) certainly
makes the proof “new enough.” Also, as I mentioned above, I find the induction
here to be exceptionally cool.
J. Michael Steele
http://www-stat.wharton.upenn.edu/∼steele/
Keywords: Cauchy-Schwarz inequality, self-generalizing inequality, mathe-
matical inequalities, mathematical olympiad
http://www-stat.wharton.upenn.edu/~steele/